The criterion of Dirichlet is a mathematical convergence criterion for series . It belongs to the group of direct criteria.
Dirichlet criterion for convergence
criteria
The series
∑
k
=
0
∞
a
k
b
k
{\ displaystyle \ sum \ limits _ {k = 0} ^ {\ infty} a_ {k} b_ {k}}
converges with if is a monotonically decreasing zero sequence and is the sequence of the partial sums
a
k
∈
R.
,
b
k
∈
C.
{\ displaystyle a_ {k} \ in \ mathbb {R}, b_ {k} \ in \ mathbb {C}}
(
a
k
)
k
∈
N
{\ displaystyle (a_ {k}) _ {k \ in \ mathbb {N}}}
(
B.
n
)
n
∈
N
{\ displaystyle (B_ {n}) _ {n \ in \ mathbb {N}}}
B.
n
=
∑
k
=
0
n
b
k
{\ displaystyle B_ {n} = \ sum \ limits _ {k = 0} ^ {n} b_ {k}}
is limited .
proof
The following applies (see partial summation )
∑
k
=
0
n
a
k
b
k
=
a
n
+
1
B.
n
+
∑
k
=
0
n
B.
k
(
a
k
-
a
k
+
1
)
{\ displaystyle \ sum \ limits _ {k = 0} ^ {n} a_ {k} b_ {k} = a_ {n + 1} B_ {n} + \ sum \ limits _ {k = 0} ^ {n } B_ {k} (a_ {k} -a_ {k + 1})}
.
The first summand converges to zero, since it is bounded by a constant according to the assumption and converges to zero. The second summand even converges absolutely , because for all and with it
B.
n
{\ displaystyle B_ {n}}
M.
{\ displaystyle M}
a
n
{\ displaystyle a_ {n}}
a
k
-
a
k
+
1
≥
0
{\ displaystyle a_ {k} -a_ {k + 1} \ geq 0}
k
{\ displaystyle k}
∑
k
=
0
n
|
B.
k
(
a
k
-
a
k
+
1
)
|
≤
∑
k
=
0
n
M.
(
a
k
-
a
k
+
1
)
=
M.
(
a
0
-
a
n
+
1
)
→
M.
a
0
{\ displaystyle \ sum \ limits _ {k = 0} ^ {n} \ left | B_ {k} (a_ {k} -a_ {k + 1}) \ right | \ leq \ sum \ limits _ {k = 0} ^ {n} M (a_ {k} -a_ {k + 1}) = M (a_ {0} -a_ {n + 1}) \ rightarrow Ma_ {0}}
.
Everything is shown.
Dirichlet criterion for uniform convergence
The series
∑
k
=
0
∞
a
k
(
x
)
b
k
(
x
)
{\ displaystyle \ sum \ limits _ {k = 0} ^ {\ infty} a_ {k} (x) b_ {k} (x)}
is uniformly convergent in the interval if there the partial sums of the series are uniformly bounded and if there the sequence converges uniformly to zero, for every fixed monotone.
J
{\ displaystyle J}
∑
b
k
(
x
)
{\ displaystyle \ textstyle \ sum b_ {k} (x)}
(
a
k
(
x
)
)
{\ displaystyle (a_ {k} (x))}
x
{\ displaystyle x}
See also
Individual evidence
↑ Harro Heuser : Textbook of Analysis . Part 1. 17th edition. Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-0777-9 , IV, sentence 33.14, p. 208/643 p .
↑ Konrad Knopp : Theory and application of the infinite series . 6th edition. Springer, Berlin / Heidelberg 1996, ISBN 3-540-59111-7 , pp. 342 ff ./604 S ( edition 1964 ( memento from January 11, 2013 in the web archive archive.today ))
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">